Categorification of the Kauffman bracket skein module of I-bundles over surfaces

Details Categorification of the Kauffman bracket skein module of I-bundles over surfaces Categorification of the Kauffman bracket skein module of I-bundles over surfaces

2004-09-21

arxiv.org/abs/math/0409414v2

Algebr. Geom. Topol. 4 (2004) 1177-1210

Mathematics

Quantum Algebra

Version 2 was obtained by merging math.QA/0403527 (now removed) with Version 1. This version is published by Algebraic and Geo

Scientific paper

Khovanov defined graded homology groups for links L in R^3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over surfaces F not D^2 (except for the homology with Z/2 coefficients only). Hence, the goal of this paper is to provide a nontrivial generalization of his method leading to homology invariants of links in M with arbitrary rings of coefficients. After proving the invariance of our homology groups under Reidemeister moves, we show that the polynomial Euler characteristics of our homology groups of L determine the coefficients of L in the standard basis of the skein module of M. Therefore, our homology groups provide a `categorification' of the Kauffman bracket skein module of M. Additionally, we prove a generalization of Viro's exact sequence for our homology groups. Finally, we show a duality theorem relating cohomology groups of any link L to the homology groups of the mirror image of L.

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